AP Statistics Ntes Uit Eight: Itrducti t Iferece Syllabus Objectives: 4.1 The studet will estimate ppulati parameters ad margis f errrs fr meas. 4.2 The studet will discuss the prperties f pit estimatrs, icludig biasedess ad variability. 4.3 The studet will discuss the lgic f cfidece itervals, the meaig f a cfidece level ad a iterval, ad prperties f cfidece itervals. Whe we select a sample, we wat t ifer sme cclusi abut the ppulati that the sample represets. I this uit, we will be itrduced t the tw mst cmm types f frmal statistical iferece: Cfidece Itervals ad Tests f Sigificace. Bth types f iferece are based the samplig distributis f statistics. The purpse f this uit is t describe the reasig used i iferece we will study specific prcedures i later uits. Statistical Iferece prvides methds fr drawig cclusis abut a ppulati frm sample data. Pit estimate A pit estimate f a ppulati characteristic is a sigle umber that is based sample data ad represets a plausible value f the characteristic. Example: A sample f weights f 34 male freshma studets was btaied. If e wated t estimate the true mea f all male freshma studets, yu might use the sample mea as a pit estimate fr the true mea. Bias A statistic with mea value equal t the value f the ppulati characteristic beig estimated is said t be a ubiased statistic. A statistic that is t ubiased is said t be biased. Origial distributi Samplig distributi f a ubiased statistic Samplig distributi f a biased statistic True value Variability Give a chice betwee several ubiased statistics that culd be used fr estimatig a ppulati characteristic, the best statistic t use is the e with the smallest variability r the smallest stadard deviati. Ubiased samplig distributi with the smallest stadard deviati, the Best chice. True value 1
Cfidece Itervals Defiiti: A cfidece iterval fr a parameter is a iterval f plausible values fr the ppulati characteristic. It is cstructed s that, with a chse degree f cfidece, the value f the parameter will be captured iside the iterval. We take the pit estimate, x, ad add ad subtract the same umber t that, ad that gives us a iterval. (estimate margi f errr) Cfidece Level: The cfidece level assciated with a cfidece iterval estimate specifies the success rate f the methd used t cstruct the iterval. The usual chices fr levels are 9%, 95% ad 99%, althugh thers are pssible. The level gives the prbability that the iterval will capture the true parameter value i repeated samples. Fr example, if the level is 95%, we are sayig that i the lg ru, 95% f the resultig itervals wuld capture the true value f the parameter beig estimated. Recall Samplig distributis Fr the samplig distributi f, x ad ad x ad the samplig distributi f is apprximately rmal whe the sample size is sufficietly large ( > 3). That meas that apprximately 95% f all large samples will result i a value f that is withi 1.96 x 1.96 f the true ppulati mea. Equivaletly, this meas that fr 95% f all pssible samples, will be i the iterval 1.96 t 1.96. Cditis fr cstructig a cfidece iterval fr : 1. The data is frm a SRS frm the ppulati f iterest. 2. The samplig distributi f x is apprximately rmal. That meas, either the ppulati is kw t be rmal, r the sample size is large eugh fr us t use the Cetral Limit Therem (CLT). 2
Syllabus Objective: 4.6 The studet will calculate the cfidece iterval fr a mea. Cstructig a cfidece iterval fr a sample mea Chse a SRS f size frm a ppulati havig ukw mea ad kw stadard deviati. A level C cfidece iterval fr is: x z. Margi f errr: z is kw as the margi f errr (m). z is the critical value that we lk up the rmal prbability table ad is the sample size. Critical value: Oce yu pick a cfidece level, yu must fid the critical value (zscre) assciated with that level. Fr example, if we wat t be 8% cfidet (C =.8), the we must fid the z-scre that separates the middle 8% frm the rest f the data. Subtract.8 frm 1, divide that by 2 ad we get.1. Usig the rmal prbability table, fid the z-scre with a prbability f.1 t the left f it. (It is 1.28 see belw). Mst cmm critical values: C = 9% Tail area =.5 z = 1.645 C = 95% Tail area =.25 z = 1.96 C = 99% Tail area =.5 z = 2.575 Geeral cstructi f a C.I.: 1. Idetify the ppulati f iterest ad the parameter yu wat t draw cclusis abut. 2. Chse the apprpriate iferece prcedure. Verify the cditis fr usig the selected prcedure. 3. If the cditis are met, carry ut the iferece prcedure. CI = estimate margi f errr. 4. Iterpret yur results i the ctext f the prblem. ***Oce yu chse, C, the cfidece level, the margi f errr fllws frm this chice. S, the margi f errr gets smaller (yur iterval gets arrwer) whe: 1) z is smaller, r yu pick a lw C, 2) gets smaller r 3) gets larger. 3
Sample size frmula If we rewrk the C.I. frmula frm the previus page, we ca use the frmula t slve fr 2 z, the sample size -. If we kw the level f cfidece ad margi f errr m we wat, we ca fid the sample size eeded fr that cfidece iterval. Graphical represetati f cfidece itervals: Twety-five samples frm the same ppulati gave these 95% cfidece itervals. I the lg ru, 95% f all samples give a iterval that ctais the true ppulati mea,. Nte abve that 24 f the 25 itervals captured the true mea (96%). 4
Cfidece Iterval fr a sample mea examples: Example 1 A certai fillig machie has a true ppulati stadard deviati f.228 uces whe used t fill catsup bttles. A radm sample f 36 6 uce bttles f catsup was selected frm the utput frm this machie ad the sample mea was 6.18 uces. Fid a 9% cfidece iterval estimate fr the true mea fills f catsup frm this machie. Sluti: We will fllw the fur steps previusly stated. First, we must state the ppulati parameter ad the ppulati f iterest: the true mea amut f catsup i the 6 uce bttles. Secdly, we must verify that we ca perfrm a 1-sample cfidece iterval. We kw, the ppulati stadard deviati, we assume ur sample is frm a SRS ad ur sample size f 36 is greater tha 3 s the CLT applies ad the samplig distributi will be apprximately rmal. Nw we ca mve t step 3 ad fid the cfidece iterval. x 6.18,.228, 36 T be 9% cfidet, the z critical value is 1.645..228 x ( z critical value) 6.18 (1.645) 6.18.63 36 The 9% cfidece iterval is: (5.955, 6.81) The last step is that we must iterpret it crrectly. Here we wuld say that we are 9% cfidet that the true mea amut f catsup i 6 uce bttles is betwee 5.955 uces ad 6.81 uces. Example 2 A maufacturer f high-resluti vide termials must ctrl the tesi the mesh f fie wires that lies behid the surface f the viewig scree. The tesi is measured by a electrical device with utput readigs i millivlts (mv). Sme variati is iheret i the prducti prcess. Careful study has shw that whe the prcess is peratig prperly, the stadard deviati f the tesi readigs is 43mV ad that the distributi is apprximately rmal. Here are the tesi readigs frm a SRS f 2 screes frm a sigle day s prducti: 269.5 297. 269.6 283.3 34.8 28.4 233.5 257.4 317.5 327.4 264.7 37.7 31. 343.3 328.1 342.6 338.8 34.1 374.6 336.1 Cstruct a 95% cfidece iterval fr the mea tesi f all the screes prduced this day. Sluti: Step 1 we idetify ur parameter ad ppulati f iterest. The ppulati f iterest is all f the vide termials prduced the day i questi. We wat t estimate, the mea tesi fr all f these screes. Step 2 We are tld that we have a SRS f the ppulati f iterest ad that the ppulati distributi is apprximately rmal, s eve thugh the sample size is less tha 3, we may prceed. All cditis are satisfied t fid the cfidece iterval. Step 3: x 36.3, 43, 2 T be 95% cfidet, the z critical value is 1.96. 5
43 x ( z critical value) 36.32 (1.96) 36.32 9.615 2 The 95% cfidece iterval is (296.75, 315.935) Based this sample, I am 95% cfidet that the true mea tesi i the etire batch f vide termials prduced that is day is betwee 296.75 mv ad 315.935 mv. Suppse a sigle cmputer scree ( = 1) with the same mea f 36.32 had bee used t fid the cfidece iterval istead f a sample f 2. Ntice hw the tw itervals cmpare belw. ****Larger samples give shrter itervals. Let s als cmpare the legths f the 9% ad 99% itervals fr this same data. ****Smaller cfidece gives shrter itervals. Example usig the sample size frmula Cmpay maagemet wats a reprt f the mea scree tesi fr the day s prducti accurate t withi 5 mv with 95% cfidece. Hw large a sample f vide mitrs must be measured t cmply with this request? Sluti: Fr 95% cfidece, the z critical value is 1.96. We als kw that 43ad the margi f errr is 5. 2 2 z (1.96)(43) 284.125, s take 285 m 5 Because, sample size, must be a whle umber, the cmpay must measure the tesi f 285 vide screes t meet maagemet s demad. 6
Syllabus Objectives: 4.9 The studet will discuss the lgic f sigificace testig, ull ad alterative hyptheses, p-values, ad e-ad tw-sided tests. 4.13 The studet will perfrm a test fr a mea. Tests f sigificace Overview We will be studyig tw types f statistical iferece. Use a cfidece iterval whe yur gal is t estimate a ppulati parameter. The secd type f iferece, called tests f sigificace, has a differet gal: t assess the evidece prvided by data abut sme claim ccerig a ppulati. The test asks: Is the result we received reasable r culd we have gtte this result simply by chace? Hyptheses The test csists f tw hypthesis statemets. The ull hypthesis, H, says that there is effect r chage i the ppulati. If the ull is true, the sample result is just chace at wrk. The ull hypthesis is always a statemet f equality. It ctais the ppulati characteristic ( ) ad the hypthesized value, which is a specific umber H : hypthesized value determied by the prblem ctext. The alterative hypthesis, H a, is the effect we suspect is true ad is the alterative t effect. The alterative ca be less tha, greater tha r t equal t the hypthesized value. The iequality frms are called e-sided alteratives ad the t equal t is called the tw-sided alterative. H : H : H : a a a Hypthesis example 1: Yu wuld like t determie if the diameters f the ball bearigs yu prduce have a mea f 6.5 cm. H : 6.5 H : 6.5 This is a example f the tw-sided alterative. a Hypthesis example 2: The studets eterig it the math prgram used t have a mea SAT quatitative scre f 525. Are the curret studets prer (as measured by the SAT quatitative scre)? H : 525 H : 525 a This is a example f the e-sided alterative. Test Statistic This is the fucti f sample data which a cclusi t reject r fail t reject H is based. The z test statistic fr a sample mea is z x. P-Value A P-Value, als called the bserved sigificace level, is a measure f icsistecy betwee the hypthesized value fr a ppulati characteristic ad the bserved sample. It is the prbability, cmputed assumig that H is true, that the bserved utcme wuld take a value as extreme r mre extreme tha that actually bserved. 7
The smaller the P-Value is, the strger is the evidece agaist H prvided by the data. The decisive value f P is called the sigificace level ad we write it as, the Greek letter alpha. If the P-value is as small as r smaller tha alpha, we say that the data are statistically sigificat at level. Sigificat des t mea imprtat. It just simply meas that it is t likely t happe just by chace. We get t chse the value f alpha. The mst cmm values f alpha are.1,.5 ad.1. H shuld be rejected if P-value. H shuld t be rejected if P-value. Steps fr the Hypthesis test STEP 1: Idetify the ppulati f iterest ad the parameter yu wat t draw cclusis abut. Yu ca d this i the statemet STEP 2: State the ull ad alterative hyptheses i symbls. Yu ca als state this i wrds t cmbie Steps 1 ad Step 2. STEP 3: Chse the apprpriate iferece prcedure. Verify the cditis fr usig the selected prcedure. That meas, ame the prcedure yu are dig ad check the assumptis. STEP 4: If the cditis are met, carry ut the iferece prcedure. -Calculate the test statistic the mechaics part! -Fid the P-value STEP 5: Make yur decisi Cmpare yur p-value t ad decide t Reject r Fail t Reject H STEP 6: Iterpret yur results i the ctext f the prblem. **Tests f sigificace assess the evidece agaist H. If the evidece is strg, we ca cfidetly reject H i favr f the alterative. Failig t fid evidece agaist H meas ly that the data are csistet with H, t that we have clear evidece that H is true. We either reject f fail t reject H. We ever use the wrd accept. Cautis befre we begi 1. The data MUST be a SRS frm the ppulati f iterest. 2. Because x is strgly iflueced by a few extreme bservatis, utliers ca have a big effect. 3. If the sample size is small ( < 3) ad the ppulati is t rmal, the iferece cat be accurately carried ut. 4. T carry ut these prcedures, yu MUST kw the stadard deviati,, f the ppulati. This urealistic requiremet reders ur frmulas f little use i statistical practice; hwever, we will lear i the ext uit what t d whe is t kw. 5. Remember there is mre errr assciated with ur iferece prcedures the just the cfidece level r sigificace level we chse, which takes care f samplig errrs. Practical difficulties, such as udercverage ad respse i a sample survey, ca cause additial errrs. 8
Fidig the sigificace level i the test If a sigificace level is t give, we will use.5. Techically, we d t eed a sigificace level. We ca iterpret ur p-value withut it ad state whether we thik it is statistically sigificat r t (small r large). T fid the p-value, we use the z test statistic ad the the rmal distributi table t lk it up. Yu must multiply yur p-value by tw if yu are dig a tw-sided test. The ame f ur Hypthesis test sigle sample test f a ppulati mea. Our test will take e f the three frms belw. Fidig the assciated p-value is shw. H : µ = hypthesized mea H A : µ < hypthesized mea x hypthesized mea P-value P Z H : µ = hypthesized mea H A : µ > hypthesized mea P-value P Z x hypthesized mea H : µ = hypthesized mea H A : µ hypthesized mea P-value 2P Z x hypthesized mea 9
Tests f Sigificace fr a sample mea examples Example 1 - Diet clas use artificial sweeteers t avid sugar. These sweeteers gradually lse their sweetess ver time ad maufacturers test fr lss f sweetess befre marketig them. The tasters scre the cla a sweetess scre f 1 t 1. The cla is the stred ad after fur mths, the tasters rate the sweetess agai usig the same scale. The bigger the differeces, the bigger the lss f sweetess. A sample f 1 traied tasters has a sample mea f.3. It is kw that the idividual tasters scres vary accrdig t a rmal distributi with 1. Is this gd evidece that the cla lst sweetess i strage? Sluti: Step 1: Idetify the ppulati f iterest ad the parameter yu wat t draw cclusis abut. the true mea lss f sweetess i the diet cla. Step 2: Write the hyptheses fr the test. This will be a e-sided test. If there is NO sweetess lss, the ur hypthesized value wuld be zer. If there is a lss, there wuld be a psitive umber. H : : H a Step 3: Chse the apprpriate iferece prcedure ad verify the cditis fr usig the prcedure. We are dig a e-sample mea hypthesis z test. We will assume we have a SRS f the ppulati f diet cla. We als are tld that the ppulati has a rmal distributi ad the ppulati stadard deviati is give. Step 4: Carry ut the iferece prcedure. Fid yur variables, the test statistic ad P-Value. 1, x.3, =1 x.3.3 z.95 1.316 1 Pz.95.1711 Step 5: Make yur decisi. Our P-Value is.1711. Sice a alpha level is t give i the prblem, we will use.5. Sice.1711 >.5, ur decisi is t Fail t Reject H. Step 6: Iterpret yur results i the ctext f the prblem. What ur results shw is that this culd easily happe just by chace. The sample mea is t far frm ur expected value. That is, 17% f all samples wuld give a mea scre as large r larger tha.3 just by chace whe the true ppulati mea is. A utcme this likely t ccur just by chace is t gd evidece agaist the ull hypthesis. These results are NOT statistically sigificat at the.5 level. 1
Example 2 - I a discussi f the educati level f the America wrkfrce, smee says, The average yug pers ca t eve balace a checkbk. The NAEP survey says that a scre f 275 r higher its quatitative test reflects the skill eeded t balace a checkbk. The NAEP radm sample f 84 yug Americas had a mea scre f x 272, a bit belw that level. Is this sample result gd evidece that the mea fr all yug me is less tha 275? The ppulati stadard deviati is 6. Sluti: Step 1: the true mea NAEP scre f all yug America me. Step 2: H : 275 ad H : 275 Step 3: We will use a e-sample z test. We assume the data cme frm a SRS frm all yug America me. Sice = 84, the cetral limit therem tells us that the samplig distributi f x will be apprximately rmal. Als, is kw, s all f the cditis are met. Step 4: 84, x 272, =6 x 272 275 3 z 1.45 6 2.7 84 P Value P( z 1.45).735 Step 5: Let.5. The ur P-Value is larger tha the sigificace level. (.735 >.5). This meas that we will Fail t Reject H. Hwever, te that if we had used a differet sigificat level, ur decisi wuld chage. If.1, the.735 <.1 ad we wuld Reject H ad say that ur data WAS statistically sigificat at the.1 level ad believe the alterative hypthesis t be true. Step 6: A mea scre as lw as 272 wuld ccur abut 7 times i 1 samples if the ppulati mea were 275. This is mdest evidece that the mea NAEP scre fr all yug Americas is less tha 275, but is t sigificat at the.5 level. There is NOT sigificat evidece t suggest the NAEP scre is less tha 275 at the.5 level. There is sigificat evidece t suggest the NAEP scre is less tha 275 at the.1 level. Example 3: Let us revisit the maufacturig example used i Cfidece Itervals Example 2. The maufacturer kws frm careful study that the prper tesi f 11
the mesh i a vide termial is 275 mv. Is there sigificat evidece at the 1% level that 275, give ur sample f 2 prvides a mea f 36.32? Sluti: Step 1: the true mea tesi f the screes prduced that day. Step 2: This test is a 2-sided test. H : 275 ad H : 275 Step 3: We will use a e-sample mea z test. We assume the data cme frm a SRS frm the ppulati f iterest. We were previusly tld that the ppulati fllws a rmal distributi. Als, is kw, s all f the cditis are met. Step 4: 2, x 36.32, =43 x 36.32 275 31.32 z 3.26 43 9.615 2 P Value 2 P( z 3.26) 2(.6).12 Step 5: The prblem states that.1. O this prblem, ur P-Value is smaller tha the sigificace level. (.12 <.1). This meas that we will Reject H. Step 6: Based this sample, there IS sigificat evidece t shw that the scree tesi fr the day s ppulati is t at the desired 275 mv level. The similarities betwee cfidece itervals ad sigificace tests A level tw-sided sigificace test rejects a hypthesis H : exactly whe the value falls utside a level 1 cfidece iterval fr. A 99% cfidece iterval wuld be related t a 2-sided test with a sigificace level f.1. A 95% cfidece iterval wuld be related t a 2-sided test with a sigificace level f.5. A 9% cfidece iterval wuld be related t a 2-sided test with a sigificace level f.1. Example: Agai we will revisit the maufacturig prblem. We will fid the 99% cfidece iterval fr the mea scree tesi. 43 x ( z critical value) 36.32 (2.576) 36.32 24.768 (281.55,331.9) 2 We are 99% cfidet that this iterval captures the true ppulati mea,. But ur hypthesized ppulati mea f 275 is t i the iterval. S, we cclude that ur ull hypthesis, 275, is implausible. Thus, we cclude that is differet frm 275. Nte that this is csistet with ur cclusi i Example 3 abve. If ur ull hypthesis had bee 29, the that value wuld have bee csistet with ur 99% cfidece iterval, we wuld have captured it ad we wuld t have bee able t reject H. 12
Syllabus Objective: 4.1 The studet will discuss the ccepts f Type I ad Type II errrs ad the ccept f pwer. Type I ad Type II errrs Whe yu set up a hypthesis test, the result is either strg supprt fr the alterate hypthesis (if the ull hypthesis is rejected) r there is t sufficiet evidece t refute the claim f the ull hypthesis (yu are stuck with it, because there is a lack f strg evidece agaist the ull hypthesis). A Type I errr is t reject the ull hypthesis whe it is actually true. A Type II errr is t fail t reject the ull hypthesis (believe it is true) whe actually it is false. Null Hypthesis Decisi True False Accept H Reject H N Errr Type I Errr Type II Errr N Errr The prbability f a Type I errr is deted by ad is called the level f sigificace f the test. The prbability f a Type II errr is deted by. Relatiship betwee the tw errrs: Geerally, with everythig else held cstat, decreasig e type f errr causes the ther t icrease. As, ad vice versa. The ly way t decrease bth types f errr simultaeusly is t icrease the sample size. N matter what decisi is reached, there is always the risk f e f these errrs. Which is wrse? Lk at the csequeces f Type I ad Type II errrs ad the idetify the largest that is tlerable fr the prblem. Emply a test prcedure that uses this maximum acceptable value f (rather tha aythig smaller) as the level f sigificace (because usig a smaller icreases ). Examples f Type I ad Type II errrs Example 1: Csider a medical test where the hyptheses are equivalet t H : the patiet has a specific disease ad H a : the patiet des t have the disease. The, a Type I errr is equivalet t a false egative (i.e., sayig the patiet des t have the disease whe i fact, he des.) A Type II errr is equivalet t a false psitive. (i.e., sayig the patiet has the disease whe, i fact, he des t.) Example 2: Whe a law firm represets a grup f peple i a class acti lawsuit ad wis that lawsuit, the firm receives a percetage f the grup's metary settlemet. That settlemet amut is based the ttal umber f peple i the grup the larger the grup ad the larger the settlemet, the mre mey the firm will receive. 13
A law firm is tryig t decide whether t represet car wers i a class acti lawsuit agaist the maufacturer f a certai make ad mdel fr a particular defect. If 5% r less f the cars f this make ad mdel have the defect, the firm will t recver its expeses. Therefre, the firm will hadle the lawsuit ly if it is cviced that mre tha 5 percet f cars f this make ad mdel have the defect. The firm plas t take a radm sample f 1, peple wh bught this car ad ask them if they experieced this defect i their cars. (a) Defie the parameter f iterest ad state the ull ad alterative hypthesis that the law firm shuld test. Sluti: p = the true prprti f all cars f the specified make ad mdel that have the defect. H : p.5ad H : p.5 (b) I the ctext f the situati, describe Type I ad Type II errrs ad describe the csequeces f each f these fr the law firm. Sluti: Type I errr: The law firm believes that the prprti f cars that have the defect is greater tha.5, whe i fact it is t. Csequece f the Type I errr: The firm will t recver its expeses, resultig i a lss t the firm. Type II errr: The law firm is t cviced that the prprti f cars that have the defect is greater tha.5, whe i fact it is. Csequece f the Type II errr: The firm will miss a pprtuity t make mey this case. Pwer Defiiti: The prbability that a fixed level sigificace test will reject H whe a particular alterative value f the parameter is true is called the pwer f the test agaist that alterative. The pwer f a test agaist ay alterative is 1 P(Type II errr) r 1 P( ). If the prbability f a Type II errr is 1% (.1), the the Pwer f the test is.9 r 9%. A high pwer is desirable. A P-Value describes what wuld happe suppsig the ull hypthesis is true. Pwer describes what wuld happe suppsig that a particular alterative is true. There are fur ways t icrease the pwer f a test. 1. Icrease. If we icrease, will decrease ad the pwer will icrease. 2. Icrease the sample size. Mre data prvides mre ifrmati ad makes the test mre pwerful. 3. Decrease. Decreasig variability will affect the shape f the curve. 4. Csider a alterative hypthesis value that is farther away frm what yu believe the true parameter t be. Chse e as far as pssible frm yur hypthesized value. Usig a sigificace test with lw pwer makes it ulikely that yu will fid a sigificat effect eve if the truth is far frm the ull hypthesis. 14
Examples frm previus AP exams Example 1: A safety grup claims that the mea speed f drivers a highway exceeds the psted speed limit f 65 miles per hur (mph). T ivestigate the safety grup's claim, which f the fllwig statemets is apprpriate? (A) (B) (C) (D) (E) The ull hypthesis is that the mea speed f drivers this highway is less tha 65 mph. The ull hypthesis is that the mea speed f drivers this highway is greater tha 65 mph. The alterative hypthesis is that the mea speed f drivers this highway is greater tha 65 mph. The alterative hypthesis is that the mea speed f drivers this highway is less tha 65 mph. The alterative hypthesis is that the mea speed f drivers this highway is greater tha r equal t 65 mph. Sluti: The aswer is C. Aswers A ad B cat be crrect because the ull hypthesis is always a statemet f equality. Sice the questi states we believe it EXCEEDS the limit, greater tha is the symbl we are lkig fr. E is t crrect because it als icludes equal t 65, which is t part f the alterative hypthesis. Example 2: A radm sample has bee take frm a ppulati. A statisticia, usig this sample, eeds t decide whether t cstruct a 9 percet cfidece iterval fr the ppulati mea r a 95 percet cfidece iterval fr the ppulati mea. Hw will these itervals differ? (A) (B) (C) (D) (E) The 9 percet cfidece iterval will t be as wide as the 95 percet cfidece iterval. The 9 percet cfidece iterval will be wider tha the 95 percet cfidece iterval. Which iterval is wider will deped hw large the sample is. Which iterval is wider will deped whether the sample is ubiased. Which iterval is wider will deped whether a z-statistic r a t-statistic is used. Sluti: The aswer is A. The cfidece level C helps determie the width f the iterval. If all ther thigs remai cstat ad the level is icreased, t be that cfidet, we must icrease ur iterval makig it wider. A lwer cfidece level results i a arrwer iterval. Example 3: A csultig statisticia reprted the results frm a learig experimet t a psychlgist. The reprt stated that e particular phase f the experimet a statistical test result yielded a p-value f.24. Based this p-value, which f the fllwig cclusis shuld the psychlgist make? (A) The test was statistically sigificat because a p-value f.24 is greater tha a sigificace level f.5. 15
(B) (C) (D) (E) The test was statistically sigificat because p = 1.24 =.76 ad this is greater tha a sigificace level f.5. The test was t statistically sigificat because 2 times.24 =.48 ad that is less tha.5. The test was t statistically sigificat because, if the ull hypthesis is true, e culd expect t get a test statistic at least as extreme as that bserved 24% f the time. The test was t statistically sigificat because, if the ull hypthesis is true, e culd expect t get a test statistic at least as extreme as that bserved 76% f the time. Sluti: The aswer is D. The P-Value f.24 is large (we usually cmpare t.5) ad we wuld fail t reject the ull hypthesis, meaig the test was t statistically sigificat. Example 4: A quality ctrl ispectr must verify whether a machie that packages sack fds is wrkig crrectly. The ispectr will radmly select a sample f packages ad weigh the amut f sack fd i each. Assume that the weights f fd i packages filled by the machie have a stadard deviati f.3 uce. A estimate f the mea amut f sack fd i each package must be reprted with 99.6 percet cfidece ad a margi f errr f mre tha.12 uce. What wuld be the miimum sample size fr the umber f packages the ispectr must select? (A) 8 (B) 15 (C) 25 (D) 52 (E) 6 Sluti: The aswer is D. Usig the sample size frmula, m.12,.3 ad the z critical value fr 99.6% cfidece is 2.88. 2 (2.88)(.3).12 whle umber, we get 52. 2 (7.2) 51.84. Rudig this t the largest Example 5: I a test f the hypthesis H : = 1 versus H a : > 1, the pwer f the test whe = 11 wuld be greatest fr which f the fllwig chices f sample size ad sigificace level? (A) = 1, =.5 (B) = 1, =.1 (C) = 2, =.5 (D) = 2, =.1 (E) It cat be determied frm the ifrmati give. Sluti: The aswer is C. Tw f the ways t icrease pwer is t icrease the sample size ad icrease the sigificace level. Aswer C has the largest sample size ad largest sigificace level. 16
Example 6: The aalysis f a radm sample f 5 husehlds i a suburb f a large city idicates that a 98 percet cfidece iterval fr the mea family icme is ($41,3, $58,63). Culd this ifrmati be used t cduct a test f the ull hypthesis a test f the hypthesis H : = 4, agaist the alterative hypthesis H a : 4, at the =.2 level f sigificace? (A) (B) (C) (D) (E) N, because the value f is t kw. N, because it is t kw whether the data are rmally distributed. N, because the etire data set is eeded t d this test. Yes, sice $4, is t ctaied i the 98 percet iterval, the ull hypthesis wuld be rejected i favr f the alterative, ad it culd be ccluded that the mea family icme is sigificatly differet frm $4, at the =.2 level. Yes, sice $4, is t ctaied i the 98 percet iterval, the ull hypthesis wuld t be rejected, ad it culd be ccluded that the mea family icme is sigificatly differet frm $4, at the =.2 level. Sluti: The aswer is E. We ca make a judgmet withut dig a hypthesis test by lkig at the iterval. The iterval des NOT ctai the hypthesized value, s it is t a plausible value. Therefre we wuld reject ad say the results were statistically sigificat. Example 7: Te studets were radmly selected frm a high schl t take part i a prgram desiged t raise their readig cmprehesi. Each studet tk a test befre ad after cmpletig the prgram. The mea f the differeces betwee the scre after the prgram ad the scre befre the prgram is 16. It was decided that all studets i the schl wuld take part i this prgram durig the ext schl year. Let A dete the mea scre after the prgram ad B dete the mea scre befre the prgram fr all studets i the schl. The 95 percet cfidece iterval estimate f the true mea differece fr all studets is (9, 23). Which f the fllwig statemets is a crrect iterpretati f this cfidece iterval? (A) A > B with prbability.95. (B) A < B with prbability.95. (C) A is arud 23 ad B is arud 9. (D) Fr ay A ad B with ( A B ) 14, the sample result is quite likely. (E) Fr ay A ad B with 9 < ( A B ) < 23, the sample result is quite likely. Sluti: The aswer is E. The cfidece level des t refer t a prbability s chices A ad B are icrrect. The iterval is just statig that we are 95% cfidet that the true parameter is ctaied i this rage, s it wuld be very likely t btai a sample result i this iterval. 17